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A145904 Square array read by antidiagonals: Hilbert transform of the Narayana numbers A001263. 5
1, 1, 1, 1, 3, 1, 1, 6, 5, 1, 1, 10, 16, 7, 1, 1, 15, 40, 31, 9, 1, 1, 21, 85, 105, 51, 11, 1, 1, 28, 161, 295, 219, 76, 13, 1, 1, 36, 280, 721, 771, 396, 106, 15, 1, 1, 45, 456, 1582, 2331, 1681, 650, 141, 17, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

Refer to A145905 for the definition of the Hilbert transform of a lower triangular array. For the Hilbert transform of A008459, the array of type B Narayana numbers, see A108625.

This seems to be a duplicate of A273350. - Alois P. Heinz, Jun 04 2016. This could probably be proved by showing that the g.f.s are the same. - N. J. A. Sloane, Jul 02 2016

LINKS

Table of n, a(n) for n=0..54.

FORMULA

Row n generating function: 1/(n+1) * 1/(1-x) * Jacobi_P(n,1,1,(1+x)/(1-x)) = N_n(x)/(1-x)^n where N_n(x) denotes the shifted Narayana polynomial N_n(x) = sum{k = 1..n} A001263(k)*x^(k-1) of degree n-1.

Conjectural column n generating function: N_n(x^2)/(1-x)^(2n+1).

The entries in row n are given by the values of a polynomial function p_n(x) at x = 0,1,2,... . The first few are p_1(x) = 2x + 1, p_2(x) = (5x^2 + 5x + 2)/2, p_3(x) = (2x + 1)*(7x^2 + 7x + 6)/6 and p_4(x) = (7x^4 + 14x^3 + 21x^2 + 14x + 4)/4. These polynomials appear to have their zeros on the line Re x = -1/2; that is, the polynomials p_n(-x) appear to satisfy a Riemann hypothesis. The corresponding result for A108625 is true (see A142995 for details).

Contribution from Paul Barry, Jan 06 2009: (Start)

The g.f. for the corresponding number triangle is:

1/(1-x-xy-x^2y/(1-x-x^2y/(1-x-xy-x^2y/(1-x-x^2y/(1-x-xy-x^2y.... (a continued fraction). (End)

This g.f. satisfies x^2*y*g^2 - (1-x-x*y)*g + 1 = 0. - R. J. Mathar, Jun 16 2016

G.f.: ((y-1)*sqrt(((x^2+2*x+1)*y-x^2+2*x-1)/(y-1))+(-x-1)*y-x+1)/(2*x^2*y). - Vladimir Kruchinin, Jan 15 2018

T(n,m) = 1/(n+1)*Sum_{i=0..m+1} C(n+1,i-1)*C(n+1,i)*C(n+m-i+1,m+1-i). - Vladimir Kruchinin, Jan 15 2018

EXAMPLE

The array begins

n\k|..0.....1.....2.....3.....4.....5

=====================================

0..|..1.....1.....1.....1.....1.....1

1..|..1.....3.....5.....7.....9....11

2..|..1.....6....16....31....51....76

3..|..1....10....40...105...219...396

4..|..1....15....85...295...771..1681

5..|..1....21...161...721..2331..6083

...

Row 2: (1 + 3x + x^2)/(1 - x)^3 = 1 + 6x + 16x^2 + 31x^3 + ... .

Row 3: (1 + 6x + 6x^2 + x^3)/(1 - x)^4 = 1 + 10x + 40x^2 + 105x^3 + ... .

MATHEMATICA

Table[1/(# + 1)*Sum[Binomial[# + 1, i - 1] Binomial[# + 1, i] Binomial[# + k - i + 1, k + 1 - i], {i, 0, k + 1}] &[m - k], {m, 0, 9}, {k, 0, m}] // Flatten (* Michael De Vlieger, Jan 15 2018 *)

PROG

(Maxima)

taylor(((y-1)*sqrt(((x^2+2*x+1)*y-x^2+2*x-1)/(y-1))+(-x-1)*y-x+1)/(2*x^2*y), x, 0, 10, y, 0, 10);

T(n, m, k):=1/(n+1)*sum(binomial(n+1, i-1)*binomial(n+1, i)*binomial(n+m-i+1, m+1-i), i, 0, m+1); /* Vladimir Kruchinin, Jan 15 2018 */

CROSSREFS

Cf. A001263, A005891 (row 2), A063490 (row 3), A108625 (Hilbert transform of h-vectors of type B associahedra).

Cf. also A273350.

Sequence in context: A085478 A123970 A055898 * A273350 A278132 A203950

Adjacent sequences:  A145901 A145902 A145903 * A145905 A145906 A145907

KEYWORD

easy,nonn,tabl

AUTHOR

Peter Bala, Oct 31 2008

STATUS

approved

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Last modified February 21 14:40 EST 2018. Contains 299414 sequences. (Running on oeis4.)